Gray's extension of Cox's proportional hazards (PH) model for right-censored survival data allows for a departure from the PH assumption via introduction of time-varying regression coefficients (TVC) using penalized splines. Gray's work focused on estimation, inference and residual analyses, but no estimator for the survival function has been proposed. We derive a survival function estimator for one important member of the class of TVC models - a piecewise-constant time-varying coefficients (PC-TVC) model. We also derive an estimate for the confidence limits of the survival function. Accuracy in estimating underlying survival times and survival quantiles is assessed for both Cox's and Gray's PC-TVC model using a simulation study featuring scenarios violating the PH assumption. Finally, an example of the estimated survival functions and the corresponding confidence limits derived from Cox's PH and Gray's PC-TVC model, respectively, is presented for a liver transplant data set.In the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right-censored survival data - additive and multiplicative models for the conditional hazard rate. A particular attention is given to data exhibiting time-varying regression coefficients. The class of multiplicative models is represented by Cox PH model and Gray's TVC model, respectively, and for additive models we use Aalen's linear model. Both Gray's TVC model and Aalen's linear model incorporate time-varying coefficients. A simulation study is performed to cross-analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate. The effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect, bias and mean square error observed for each conditional model-based estimator of survival. We also show that Aalen's model formulae is a first order Taylor series approximation of that of Gray's model which explains the comparably higher flexibility on part of the Aalen's model as compared to the Cox PH when the Gray's TVC model for the data is misspecified.
